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Why Arrow’s theorem is a scam.

Since I seem to be on a bit of a social choice kick, let me expound a bit on my pet peeve in social choice theory, the misuse of Arrow’s theorem.

The usual statement of Arrow’s theorem goes something like this: Any voting system which is Pareto efficient and satisfies independence of irrelevant alternatives is a dictatorship.

Pareto efficiency is an extremely weak condition. It simply says that if everyone who votes prefers A to B, B cannot win (certainly any reasonable voting system satisfies this).

Independence of irrelevant alternatives is more touchy. A more accurate name might be the “impossibility of spoilers.” It requires that adding new choices (i.e. candidates) should not change the relative finishing position of the original choices.

IIA is a bit difficult to think about because no voting system you are likely to have heard of satisfies it (by Arrow’s theorem). For example, in the American presidential election, adding a new candidate (let’s call him N) could easily change the outcome between two other candidates (which one might call B and G), for example if G would win a two-way vote with B, but if N runs, he will get the votes of more people who prefer G to B, he can throw the election to B (not that I’m claiming anything like this has ever happened).

Now, spoilers in elections are an upsetting phenomenon (especially for Democrats over the last 8 years), and it would really be nice if we could avoid them. Unfortunately, Arrow’s theorem says that we can’t. Right?

Wrong!

Here’s an examples of a voting system which seems to conflict with Arrow’s theorem:

Rank voting. In this system, each voter gives each candidate a numerical score on some scale (say 1-10). The voter’s scores for each candidate are summed, and the candidate with the highest sum (alternatively, average) wins.

One particularly simple version of this is approval voting, where the options are “approval” (1) or “disapproval” (0).

This system is obviously Pareto efficient and satisfies IIA (changing the score of any candidate won’t affect the relative placement of two others). So how can it be consistent with Arrow’s theorem?

Well, implicit in Arrow’s theorem is a definition of what a “voting system” is, and Arrow’s theorem is based on a rather restrictive (some might say “stupid”) choice. The only acceptable input of a voting system in the Arrow’s theorem schema is an ordered list with no way of measuring the intensities of support for a candidate. So, as far as Arrow is concerned, approval voting isn’t a voting system.

So, give a moment of thought to that the next time somebody claims, well, pretty much anything is a consequence of Arrow’s theorem. And for God’s sake, promote approval voting. It’s sick that people are so excited about instant runoff voting, when approval voting is so obviously superior.

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49 thoughts on “Why Arrow’s theorem is a scam.”

Such a voting system adds some nuance to any particular individual’s voting strategy. Suppose that there are three candidates, named O, C, and M. Suppose further that I have a slight preference for O over C, and an enormous preference for C over M. If I vote according to the spirit of approval voting, I should probably vote 1,1,0; this is also the best strategy to minimize the probability of the ghastly event that M should win. But If I’m confident that M won’t win, I’m more likely to get what I want by voting 1,0,0. Which should I do?

This came up in practice recently during my school’s hiring meetings. We had to vote for who we want to invite or hire, with a maximum of one vote per candidate, but a maximum of n>1 votes in total (with n varying at different stages of the process). If I wanted to hire an algebraic geometer and I especially wanted to hire Grothendieck, I would have been faced with the unpleasant choice between voting for both Deligne and Grothendieck, or just for Grothendieck.

It’s true that approval voting isn’t perfect, but it still has a big advantage over the preference rank voting systems, in that it’s immune to the Gibbard-Satterthwaite theorem. While some choices of how many candidates to approve result in better results than others, at least it’s never in your advantage to lie about your preferences.

Ben. Sure it’s in your advantage to lie about your preferences. Even if you really might approve of more than one candidate, you are hurting your favorite choice by approving of any other candidate than that one. So people tend not to do it. Approval voting moves toward bullet voting, and, basically, plurality voting.

Imagine approval voting in this year’s multi-candidates primaries. The campaigns all would be telling their core supporters to only vote for their candidate, and disciplined voters would do just that. The naive, “can’t we all just get along” voters would approve of more than one candidate and the result is a big edge for the disciplined voters’ favorite candidate.

Systems where lesser choices hurt higher choices have very little track record of success for these reasons. I would be interested in seeing more uses of it to make it clear this would regularly happen, but it’s very hard to get any hotly contested elections run with approval voting — its advocates would do well to try rather than spend so much time bashing other reforms that have a track record. Instant runoff voting has a much better history of being used because it makes sense to folks — and has a long history of working. See http://www.fairvote.org/irv.

It’s true that in approval voting, one never is induced to actually lie about one’s preferences, but it is still possible that one’s voting pattern is not linearly proportional to the strength of one’s preferences. This can still lead to significant distortion.

Anonymous’s example is a case in point. Suppose that 31% of the electorate is like Anonymous; preferring O to C by a slight margin, and C to M by a huge margin. 29% prefer C to O by a slight margin, and O to M by a huge margin. The remaining 40% prefer M to either of C and O by a huge margin (and, let’s say, prefer O to C by a slight margin, though it’s not too relevant for this discussion). The voters in the first two camps may well think that the election is sure to go to either O or C (who are acceptable to 60% of the population, whereas M is only acceptable to 40%), and will therefore maximise the power of their vote by voting 1,0,0 or 0,1,0. Meanwhile, the third camp solidly votes 0,0,1, and thus M wins despite having a 60% disapproval rating. In contrast, IRV would eliminate C and then lead O to win the final election, being the first choice of 31% and an acceptable second choice of 29% of the electorate.

It is true that no system is perfect, but selecting a system solely because it avoids one type of problem is a little dangerous, as it may unwittingly increase the chance of susceptibility to other sorts of problems. (But either approval or IRV would definitely be superior to a first-past-the-post system.)

I’ll just note that I wasn’t really intending to make the post above about the benefits of IRV vs. approval voting. In part, I think approval voting actually does need some more real world testing before we know how well it works (as with IRV). Like, Terry, I think either would be a great alternative to the current first-past-the-post system in the US (I also think that replacing single member districts with some more proportional alternative like the NZ or Irish system would make a much bigger difference than changing voting systems, and would advocate for that before any particular voting system).

In part, I think it also depends a lot on what your personal views on what sort of fairness is important in an election. IRV takes the first preferences of voters very seriously, whereas I hope that approval voting could open the door for a more consensus model for winning an election. After all, Rob’s criticism w.r.t. primaries could be phrased as “The candidate that does the best job of convincing other candidates’ supporters that they are an acceptable alternative will win” which doesn’t sound so bad to me. I’m also perhaps more skeptical about how good people are at constructing consistent preference lists.

As to the degree that people really will bullet vote in any given election in any given system, that’s a question of social psychology, not mathematics (is there any actual evidence on this point? There’s nothing linked from the wikipedia article). I can attest that had I been given the option, I would have voted for 3 candidates in this year’s primary (if you want to know which 3, you’ll just have to email me. I’ll try to keep this blog apolitical as long as I can), rather than 1, but, of course, I’m not exactly the typical voter.

I forgot, in response to Rob’s first point, I don’t consider that lying. I suppose this is a disputable point, but it certainly is true that it’s never advantageous to say you prefer candidate A to candidate B, when the reverse is true (which is not by any means a claim IRV can make).

An interpretation of Arrow’s theorem I’ve found appealing is that a voting system isn’t determining “the voters’ true choice” but rather “the candidate most likely to be appropriate for the population given the information collected,” and the reason that irrelevant alternatives can change the latter is that they add to the “information collected.”

An example. We have A, B, C.

A vs B: Suppose 55% prefer A to B. Given only this information, A is clearly be the candidate most likely to be appropriate.

A vs B vs C: Suppose C enters the race, and 60% prefer B to C, and 60% prefer C to A. This is a bit funny, and we have a cycle in the (directed) graph of pairwise preferences. It’s now not as clear who is the candidate most likely to be appropriate given only this info, but essentially any reasonable method of determining this arrives at the answer of B. (Disclaimer: If we add in voters’ ordered lists of preferences so we see who was most often voters’ first choice, we have yet more info, and perhaps B won’t as clearly be correct.)

An appealing voting method (or rather collection of voting methods) in line with the above philiosophy is the “Condorcet method” (see http://en.wikipedia.org/wiki/Condorcet_method). It’s still susceptible to tactical voting, but lacks several flaws that IRV has. Perhaps its main flaw is that it only takes into account pairwise preferences.

Since you mentioned New Zealand, I thought I’d chip in, since I live there. I think you were referring to our Mixed-Member Proportional (MMP) system of representation in Parliament. (We have only one house of Parliament, rather than a Lower House together with a Senate or House of Lords or whatever.)

In MMP (which I think is roughly the system used in Germany and Italy), the country is still divided into several geographical electorates, each of which elects one Member of Parliament (MP), using an ordinary first-past-the-post (FPP) election (the electorate vote). Each party also produces a ranked list of candidates, and voters can vote for whichever party they prefer (the party vote). Then, the number of MPs is topped up to 120 by candidates from the parties’ lists. The top-ups are given to the parties in such a way that the total number of MPs from each party is roughly proportional to the party vote they received.

Yes, it does make Parliament more proportional in a way, but on the other hand, it entrenches the party system, and all of the us-against-them division that goes with it. Even though people can vote for anyone at all as their local MP, knowing that they won’t disadvantage their preferred party, not a single independent MP has been elected in any of the four MMP elections we’ve had here since 1996; all the publicity is about parties, so that’s all that most people bother to think of when they go to vote.

To add to the confusion, the elected members of our District Health Boards (DHBs) are all elected using Single Transferable Vote (STV), which is like IRV, but generalized to a multi-winner election. The mayoral and city council elections (which are held at the same time as the DHB elections) vary throughout the country between FPP and STV.

The first time STV was used here for those elections, there were significant delays in announcing the results in some areas (weeks-long in one place, if I remember correctly). This is partly because the contract for counting the votes was given to the lowest bidders in each area (which is an appalling idea, by the way). Also, though, it’s because STV is a fundamentally slow voting system to count. You have to keep track of each possible ranking of candidates.

In the most recent DHB election, I had a list of 22 (I think) candidates to rank, seven of which would win. Now, 22! is a very large number—much larger than the number of voters. So, effectively, the vote-counters had to keep track of each individual vote—you can’t distribute the counting and just send totals to the centre that does the final tally. Also, the problem that voters have of choosing among 22! possible rankings isn’t at all trivial—an optimal search among the space of possible rankings requires 69 or 70 yes/no questions. Even if you only want to rank your top 7, you’ve got to choose from about 2^30 options. Fortunately, we’re not obliged to rank all the candidates, but the only way to express “I want anyone but Z” in STV is to rank everyone else above Z.

Another problem with STV is that it only looks at the top of your ranking. Your vote only ever descends down your preference list if your first preference is knocked out (in which case your whole vote is passed on to your next preference) or elected (in which case the “spare” part of your vote that they no longer need is inherited by your next preference). This means that if the overall highest-ranked runner-up was ranked highly on your personal preference list, then most of your preferences (which you spent hours deciding about) are ignored.

Condorcet voting, though, can improve on STV. For a start, it takes account of all of your pairwise preferences. Secondly, counting can be distributed: all that matters is the number of voters who prefer A to B and the number of voters who prefer B to A, for each pair of candidates, (A, B). The final ranking needs to be done centrally, as always, but the details of individual votes don’t matter for that. Also, in the most general form of Condorcet voting, there’s no need to force people to have a ranked list. They can express a preference for A over B, and a preference for C over D, without expressing any other preference.

A common objection to STV in New Zealand is that it’s too complicated for many voters to understand. This would also apply to Condorcet voting, even more so to any attempt to implement the most general form of it. However, I’m not convinced that it’s actually a disadvantage if a voting system requires the voters to exercise some intelligence in order to cast a valid vote.

Now I’ll drift briefly to something slightly more on-topic. As has already been pointed out, although approval voting doesn’t encourage people to lie about the order of their preferences, it almost unavoidably encourages them to lie about the magnitude of their preferences. I suspect that this is a problem with any voting system that purports to take account of such magnitudes. Voting systems that ignore magnitudes of preferences avoid this problem, but then, as you say, they’re subject to Arrow’s theorem. A good question, then, is how accurate the polls need to be to encourage people to lie about their preferences in, say, Condorcet voting.

It seems that the problem we encounter with anonymous’s example (with Tao’s numbers) is that on a coarse-grained scale, two candidates are essentially indistinguishable to the electorate, and should only be separated at some later stage in the decision process – if they aren’t eliminated first. IRV does a first approximation to this, but it would be nice if we could quantify it in some holistic way.

Here’s a (vague and potentially fatally flawed) attempt: Label the axes of R^n with the candidates, and have voters pick positive reals according to how much they like someone. Rescaling puts each voter’s preferences on an n-1 simplex. We can’t just take an average and see what Voronoi cell it falls in, since that fails the discussed example (M wins – this is just a continuum version of approval voting). Instead, we muck around with the metric on the simplex, to collapse some subsimplices where voters are most clustered near the barycenter under some projection. If we repeat this enough times, we get a 1-simplex, and choose to eliminate the less popular of its boundary points. Then repeat with the preimage of the non-eliminated point, until one candidate remains.

Tim McKenzie has a good point at the end of his comment, about ordering vs. magnitude of preferences. Arrow was well aware that he was looking only at a restricted notion of voting, but he did it deliberately for philosophical reasons. Relative preferences are easily compared between people, but strengths of preferences are not; it’s not easy to compare utility functions between people. In particular, for approval voting there is no principled way to characterize what “approval” should mean for different people (it’s vague and ill-defined). One could try to patch it up, for example by measuring utility according to how much one is willing to pay to get one’s way, but that would of course be grossly unfair in other ways. In the end, it’s not clear how much attention society should even pay to strength of preferences: if I desperately want my candidate to win, should that outweigh a dozen people’s mild preferences? It’s hard to say. Using just the voters’ linear orderings of the candidates is awfully restrictive, but it makes some sense philosophically. In the real world, we can do better.

Approval voting seems great to me. It has its problems, but it is overwhelmingly better than IRV (let alone plurality voting). Aside from the obvious objection that it’s absurd to privilege first-place votes over all others, I don’t like IRV’s complicated dynamics. I don’t think anybody really understands at a deep level how it works, or can give a good argument for why it selects the right candidate. It’s just an ad hoc attempt to patch up plurality voting to keep spoiler candidates from ruining elections between two strong candidates. It works fine for doing that, but once you have numerous strong candidates, the question of who will win becomes quite complex. Tactical voting is bad enough, but it’s even worse when it’s hard to guess what the effect of your vote will actually be!

For me, one crucial aspect of a good voting system is the ability to explain what happened and give a compelling justification of the result. There’s no simpler or more compelling justification than “more voters approved of X than of any other candidate.” By contrast, talking about first-place votes is already suspect, and when you start a multi-round process of transferring them between candidates, I have no faith that the final result means anything important.

It drives me nuts that IRV has somehow assumed the mantle of successor to plurality voting, when its only qualification is that it looks comfortably familiar. I’ve actually had conversations with relatives who have complained that approval voting violates the sacred rule of “one person, one vote”, as if that somehow made it unfair, so I guess it’s a tougher sell. Still, I’ve never understood what makes lots of activists (like the FairVote people) get behind such a ludicrous system.

Anonymous 2 [note- the two anonymous commenters are different people]-

I fear the fact that it “feels wrong” to vote for as many candidates as you want, while completely unjustified from a mathematical perspective, probably dooms the method to be a niche choice. Oh well.

I’ll also concur that the dynamics behind IRV are a little unnerving (IRV advocates’ untroubled acceptance of non-monotonicity, in particular, feel troubling to me, as a mathematician), though I think it’s enough of an improvement over plurality voting that I’d be basically happy to see it installed nation-wide (though my mind would be easy to change if problems start showing up in municipal implementations of it).

This is getting away from the math a bit, but let me assure you that I’ve thought about the increase in party discipline that follows from list systems, and my general feeling is that it’s a worthwhile trade-off, for a couple of reasons:

-it’s basically too much to ask a voter to figure out the positions of their local candidates AND the national party and then figure out how to balance the differences between these. If you basically only have to vote for a party, you at least know where the party stands. (This has been a serious problem in the US in past, for example with respect to civil rights in the 60’s, when voting for a pro-civil rights Democrat in the North would entrench the segregationalist Democratic committee chairs in Congress.)

By the way, approval voting isn’t Pareto efficient. Suppose everyone has a mild preference for A over B, but no individual vote distinguishes them (that is, everyone who gives 1 to A also gives 1 to B). If you need a single winner, and A and B are equal-top scorers in the election, then the system must make an arbitrary choice between the two. If it chooses B, then Pareto efficiency is violated. Of course, such a situation is unlikely, but so is any election at all where all the voters (including, presumably, the candidates themselves) prefer one candidate over another. The problem is getting a system to simultaneously satisfy Pareto efficiency and IIA in all possible cases.

If you’re willing to support IRV over FPP/plurality voting, you should probably support Condorcet voting over them both.

A Condorcet winner is a candidate who would win a one-on-one election against every other candidate. The Condorcet method will always elect a Condorcet winner if there is one. IRV won’t. FPP can even elect a Condorcet loser. A problematic example for IRV is this, from condorcet.org:
38 ABC
7 CAB
15 BCA
15 BAC
25 CBA
B is a Condorcet winner (55 of 100 voters prefer B over A and 68 prefer B over C), but IRV eliminates B first, and elects A.

Also, as I’ve already mentioned, counting votes in Condorcet elections is much quicker than in IRV elections.

And briefly, about MMP again: in New Zealand the situation is really quite bad. MPs don’t actually need to be present in Parliament’s debating chamber to have their vote cast by their party’s “whip”. In fact, I think parties can even force their MPs to vote a particular way; “conscience” votes, where a party lets its MPs vote individually, are quite rare. If we can’t expect voters to be able to think about both their local candidates’ and the various parties’ policies, then I’d prefer that they thought about their local candidates’ policies, and that the MPs always voted according to their consciences.

And bribing 61 individual MPs without getting caught is much more difficult than bribing one or two people in the party’s leadership.

Todd- that is an alternative, in the sense of another thing that you could do, but not one which would make me feel like I understood approval voting better. As I mentioned above, my concerns are primarily those of social psychology, not mathematics.

If you’re curious what’s in the literature, you might want to try following the links from the approval voting and IRV wikipedia pages. It looks like there might be data in some of them.

The post really refers more to Gibbard-Satterthwaite Theorem where a single winner has to be selected than to Arrow’s theorem where an order relation of the candidates has to be the outcome. But the two theorems are closely related. The nice thing about Arrow’s theorem and its relative that it gives the feeling that there must be a way around it. The reality is that there isn’t really a way around it. There are several nice methods like approval voting and various people found various reasons to prefer some other voting rules and regarded it as the “solution” to Arrow’s theorem. (Or the Gibbard-Satterthwaite’s theorem.) But non of these solutions are really convincing. Of course, some very strong interpretations of Arrow’s theorem that people try to promote from time to time arn’t convincing either.

I don’t have a full mathematical defense of approval voting, but somehow I like it. It is a very simple system and its strengths are attained by elegance. By contrast the philosophy of IRV is to get the desired behavior with higher-order epicycles. Another reason that I like approval voting is that it is a centrist system that dampens the incentive for negative campaigning. But I take Terry’s point that it is glib to declare one system superior just because of one or two specific advantages. There is a science of voting and you shouldn’t jump to conclusions after learning only some of its findings.

By the same token, it irritates me when an important, neglected topic such as voting systems is marketed and distorted to fit a specific ideology. Consider the site fairvote.org that promotes a system that it calls “choice voting”. What they mean is that IRV is the only good solution. Either you want IRV, or you are for an unfair vote that lacks choices. Some of the inspiration for this movement in the US is specifically the 2000 election; it’s the people who wanted to vote that Nader is best and Gore is a distant second. For them, it wouldn’t do to simply approve of both of them.

Now IRV might be a good system, but the empowerment of fringe voting should not be the driving goal. Certainly in practice, the complexity of IRV comes at a real cost. In the abstract, of course, the rules are simple enough for a computer tabulation. The problem is that it is harder to read an incomplete tally in IRV rules than, for instance, in approval voting. The polity often wants to know what is going on before the tally is finished, because some of the votes may be contested, late, sealed, etc.. Again, I don’t know all sides of this topic, but the zealous promotion of IRV makes me wonder if it is to election problems what marijuana is to pain relief: a very fashionable, very progressive solution, but not the best one.

That said, I realized that the rules for approval voting as I understand them give the political center a monopoly in a multi-seat election. AV works well in the single-seat case, but not otherwise. Does AV have a clean generalization that converges to proportional representation in multi-seat elections?

One way to interpret Arrow’s theorem/Gibbard-Satterthwaite’s theorem is that we may want to make more assumptions when analyzing a voting system for actual implementation. I’m inclined to assume the existence of a Condorcet winner. Which of the bad properties of Condorcet methods remain, if any? Is there data to suggest that this assumption is unjustified?

Can this be proved from a lesser assumption? Or perhaps just with probability one as population tends to infinity?

Since we’re talking partly about social psychology, perhaps my first experience of voting is relevant. In my first full year at school, my class had the chance to elect a few of the pupils to have the privilege of performing certain minor, but not unpleasant duties. I can’t remember what they were. The voting wasn’t at all by secret ballot—the teacher announced the names of the candidates one by one, and counted the number of hands that were raised in support. I was chastised for trying to vote for more than one person for a particular duty. When it was explained to me that you only got to vote for one person for each duty, it seemed quite arbitrary to me.

Am I unusual? Or do most children instinctively prefer approval voting until they’re taught it’s “wrong”? Possibly both.

Actually, I still think it’s a bit arbitrary, especially when, as with my city council, I see multiple-winner elections, where we do get to cast multiple votes. In general, we get to cast up to n votes for an n-winner election, which at first appears not to be arbitrary (and I certainly wouldn’t like to be restricted to voting for fewer than n candidates), but if you think about it too much, you start to wonder why we don’t just have approval voting.

Greg’s point that centrist candidates might have an advantage in AV elections is an interesting one. I think I’ve seen some commentary suggesting that Condorcet voting also has a bias towards centrists. This is only really a “problem” in multiple-winner elections, where you might want proportional representation; in a single-winner election, it’s hard to argue against electing a Condorcet winner, and if there is no Condorcet winner, then it’s unclear that the word centrist is even meaningful.

At the top of http://www.gentoo.org/news/en/gwn/20070924-newsletter.xml there’s an example of a real-word multiple-winner Condorcet election, including a link to a detailed breakdown of how the vote-counting went. In this case, they just found a winner by the Condorcet method, removed the winner, found the next winner by the Condorcet method, and so on. Since it probably wasn’t a very “political” election, but more one about experience, skill, and respect, there wasn’t really any worry about bias towards “centrism”.

My dream is of a multiple-winner Condorcet election where voters rank not the individual candidates, but entire potential councils. Of course, this massively increases the complexity of voting, to the point where it’s unlikely ever to get the consent of any non-mathematician. And voters would need to be able to put arbitrary partial orders on all of the n-choose-m possible councils (where n is the number of candidates, and m is the number of winners needed), by giving a list of instructions, in order of priority, like this:
1. Prefer councils that include more members from {A, B, C}.
2. Subject to the above, prefer councils that exclude Z.
3. Subject to the above, prefer councils that include at least one member of {D, E, F}.
And so on. Voters could still give just a ranked list of candidates (A, B, C, D, …), which would be interpreted as:
1. Prefer councils that include A.
2. Subject to the above, prefer councils that include B.
3. Subject to the above, prefer councils that include C.
…
I think this might go some way to solving the proportionality problem.

Instead of computer simulations, it would be better to do real-life experiments. I don’t see a way to simulate the human factors of politics with computers. Maybe you could represent the voters with empirical statistical models, but how could you model the politicians and election officials?

Of course in the case of IRV, it’s not a hypothetical, since it is used in Australia and New Zealand. I don’t know how often approval voting has been tried, or with what results.

Although I feel attracted to approval voting, I have to say that the Gibbard-Satterthwaite is debatable as a theoretical argument in its favor. AV is immune to this theorem only by defining it away. It overtly grants the tactical voting that the theorem says must exist. I think that that could be a good thing for a non-mathematical reason: it’s more straightforward. But I’m not sure that it would be any real victory in a world of fully rational voters.

On the other hand, I’m suspicious of the assumption of fully rational actors, not only in voting but also in puzzles like the blue-eyed islanders. In real life, of course, even highly intelligent agents only have a bounded amount of capacity to second-guess each other. I have the feeling that these bounds change the properties of voting systems and change the answers to some of these puzzles. But I haven’t thought of a rigorous formulation of any such bounds to make that clear.

One further nice thing about the Arrow impossibility theorem is that it generalizes to other contexts – not only is it impossible to construct a linear ordering from a set of linear orderings while maintaining those conditions, but the same is true for truth-assignments in place of linear orderings (this has relevance for legislation or any other group decision-making procedure), or probability functions (when aggregating expert advice), or a variety of other structures. In most cases there’s something like an IIA assumption that is clearly problematic in some way, and I think in a few cases they’ve managed to weaken that assumption in various ways and prove related possibility theorems.

I found a tendentious explanation at fairvote.org of why IRV is superior to all other voting protocols, even for single-seat elections. Their example for approval voting is particularly lame. They said that if there are two candidates and 99% of the voters approve of both of them, then the other 1% could “strategically” vote for just the one that they like better. Of course, if there are only two choices, then AV is equivalent to majority vote with abstentions, and there is nothing undemocratic about it.

But this example, and Terry’s, led me to consider the correct precepts of choosing a voting system, and the correct lesson of Arrow’s theorem. As I see it, the message of Arrow’s theorem and Gibbard-Satterthwaite is that if a democracy is faced with only three or more choices, then strategic voting is inevitable. By contrast, if there are only two choices, then majority rule is a self-consistent answer. Now usually in these discussions, a clean voting system such as majority rule is viewed as good and strategy is viewed as bad, but in many cases that’s an oversimplification. After all, if dictatorship of the majority is the answer, then it sucks to be in a minority! In fact it’s a very common story in politics that embattled minorities try to upend democracy (with violence, secession, or legal recourse) to protect their interests.

Fair Vote argues that if a majority of voters agree on their first choice, then any sound voting system would grant them that choice. But that isn’t always true. For instance, suppose that Harvard and Yale students vote on where to play the Harvard-Yale game, and the three options are (1) Harvard always, (2) Yale always, or (3) alternate games. There are about 20% more Harvard undergraduates than Yale undergraduates. In the IRV system, it is easy for the 55% majority to impose its will and get all of the games at Harvard. But in approval voting, a small fraction of the Harvard students could endorse the compromise. In this case, IRV works to destroy compromise, indeed even just the weak Condorcet criterion that a first-choice majority should always win.

It seems to me that the right analysis is to pick the system with the best game-theoretic properties, rather than the system that looks like the best abstract mapping from preferences to outcomes. It’s a mistake to view strategic voting as a monstrosity, among other reasons because the theorems say that it is intrinsic to democracy. They aren’t just saying that every voting system admits artificial extremes. Again, I don’t really know that AV is a good system, but I am drawn to it, because the examples so far come with a story that the voters could have “played it better”. You could describe Terry’s example like this: The backers of two moderate candidates refused to admit to their common insterests and instead let a radical win.

By contrast, there are perverse game-theoretic possibilities in IRV, because it is after all a type of run-off system. For example, suppose hypothetically that 45% of the voters in 2000 preferred Bush to either Gore or Nader, 30% preferred Gore to Bush and Bush to Nader, and the other 25% preferred Nader to Gore and Gore to Bush. Then 6% of the Bush voters could strategically defect to Nader, to eliminate Gore and get Bush elected. Unlike the example with Harvard and Yale, the particular kind of strategic voting is malovelent (and not just because of the specific people involved). In this case, an organized minority can destroy compromise unilaterally.

Still, the fact that IRV has a multiseat generalization that converges to proportional representation is important. Again, I would be interested to see such a generalization of AV, or know that there isn’t a reasonable one. For concreteness, suppose that in a multiseat election, the voters just submit approval lists, in only one round and in parallel. Is there a way to tally those lists so that, if the voters follow their incentives, the result converge to PR as the number of seats goes to infinity?

Thanks for that link, Greg. FairVote’s reasoning doesn’t really seem that convincing. I was especially amused by their argument that Condorcet voting is too difficult to count. In comparison to plurality voting, yes, it is difficult, but aren’t we meant to be comparing it to IRV?

Your point that majority-rule can lead to oppression of minorities is an important one. I don’t think any voting system can possibly solve that; you need some other way of protecting individual rights from arbitrary confiscation by majorities. That way might involve constitutional protection, or cultural expectations, or something else.

Anyway, I’ve had a brief first-think about generalizing AV to multi-winner proportional representation. My first instinct is to try some variation on the Sainte-Laguë or D’Hondt methods used to achieve proportionality in party list systems. So in counting votes to award the k-th seat, your approvals of remaining candidates count as 1/(s+1) of a vote each, where s is the number of candidates you voted for who’ve already won. That’s a generalization of the D’Hondt method—for the Sainte-Laguë method (which is used in New Zealand for MMP), you’d replace 1/(s+1) with 1/(2s+1). I have no idea what the advantages of each are.

I haven’t checked that this would actually converge to PR, but my instinct tells me that if it works for pre-defined lists, then it should work when voters provide their own (unordered) lists.

This shares a couple of disadvantages with IRV. First, you need to keep track of individual votes, rather than just sending totals from each voting booth to the centre where the final tally is done. I don’t know if it’s possible to avoid this without having pre-defined lists to choose from. On the other hand, with n candidates, there are only 2^n possible distinct votes, rather than the n! that IRV has, even if voters are obliged to rank the whole list, as I think they are (or at least have been in the past) in Australia.

The second problem is that it encourages people not to vote for popular candidates. If A is the most popular candidate, and if you’re confident that A will win, then you can make your votes for other candidates worth more by not voting for A, even if you support them.

Your point that majority-rule can lead to oppression of minorities is an important one. I don’t think any voting system can possibly solve that; you need some other way of protecting individual rights from arbitrary confiscation by majorities.

Certainly if any issue is taken in isolation, then history suggests that that is correct. And what I have in mind is not “individual” rights, but an arbitrary objective of any minority faction in any voting system. The minority can protect its interests by bundling its objectives with those of other factions. If enough issues are tied together, it can create a benevolent stalemate that prevents majority tyranny. Sometimes this stalemate is expressed with constitutional guarantees.

But when a minority is obsessed with one issue, or a few parallel issues, that can’t be balanced, then it typically feels cheated by the majority preference. Terrorism and secession are not uncommon outcomes. For instance the Confederacy was a minority faction that tried to break away: continuation of slavery was all-important and the democratic system ran out of options for reciprocation and compromise. Another example is Quebec’s chronic temptation to secede from Canada. Part of the point is that the embattled minority is sometimes good, sometimes evil, and sometimes neither.

Except in a few cases of an evil minority interest, I would want a voting system that facilitates compromises. Obviously in practice there is a lot of bundling; it seems necessary for a viable democracy. (And although I didn’t consider it before, it could be another reason that direct democracy implemented with a thousand yes-no referenda is a bad idea.)

One variation of approval voting which I found handy (for the task of speaker’s selection at conferences) when the number of voters is comparable to or even smaller than the number of candidates is this:

Every voter assign every candidate a polynomial in t with 0-1 coefficients. These polynomials are then added up and t is regarded as positive infinitesimal. (In practice, we used only 0,1,1+t and 1+t+t^2.) So t and its powers are only used as tie brakers.

I think approval voting is a good method when approving/not approving is a good way to model the individual original preferences regarding the candidates. Namely, when you do not need to translate a more complicated individual preferences (like a total ordering, or a real utility function) into the coarser approve/not approve. Similarly this method with polynomials is good in cases where individual preferences on alternatives are totally not Arcemedian so modeling with utility functions described by polynonials with 0-1 coefficients is realistic to start with.

Actually in cases when a sublist from the list of all candidates have to be chosen (and not just a single winner) , it can also be useful to allow voters to specify preferences of the form “Among A, B, and C, I approve at most two”.

Gil: That sounds like quite a clever method of voting for certain situations. However, I’m not sure that I fully understand your comment about approving at most two of a set of candidates. Can you explain that bit in more detail, please?

You said “rank voting” when I think you meant “range” voting. Rating candidates on a scale like 0-10 or 1-5 is range voting, approval voting being the simplest “0-1” form.

Rob Richie is doing his typical con job here when he says Approval Voting will lead to bullet voting. Does he really believe Nader voters in 2000 would have been scared to vote for a second candidate like Gore out of fear that Gore (their second choice) might then beat their first choice Nader? Well, 90% of voters who supported Nader did _not_ vote for him, so they prove Rob is wrong. But with Approval Voting they could have safely voted for Nader.

Tim, lets say you have to select 4 speakers from a list of 15.
A voter has the following considerations:

1) There are 8 of the 15 speakers out of the 15 that he approves.
2) There are 2 areas and he does not want that all 4 speakers will be from the same area.
Condition 2) is not something a voter can specify in a usual approval voting method. We let people give such instructions, and when we “count the votes” we first ranked the candidates according to the overall approving votes, and then apply the additional instructions of voters for candidates according to this initial approval ranking. (So if a voter approved 3 candidates but also asked that at most 1 will be selected, after the approval ranking was made we deleted one vote given by this voter from the two among the three with lowests ranking.)

I do not know if this is a good method. Maybe it is better that the individual voters will just approve/not approve and each one will include all his other considerations in his approval decision.

(In practice, while using polynomials for the fine tuning worked nicely as far as I remember this possibility for extra instructions was hardly used.)

There are better proportional representation systems than the archaic STV, which degrade to good single-winner voting methods, unlike STV which degrades to the rather horrible IRV in the single-winner case.

The Bayesian regret simulations by Warren Smith, for instance, use millions of random elections with multiple variables, simulating electorates with 100% honest voters all the way to 100% strategic voters, and every mixture in between. All sorts of chaos is added to simulate the complexities of human behavior, and there is a uniformity in the results in that Range Voting surpasses the other methods in ALL 720 combinations of parameters (whereas IRV is one of the worst).http://rangevoting.org/BayRegDum.html

Maybe you could represent the voters with empirical statistical models, but how could you model the politicians and election officials?

There’s real world data that can tell you plenty about the kinds of rates of error in voting with, and counting, various types of voting systems.

Of course in the case of IRV, it’s not a hypothetical, since it is used in Australia and New Zealand. I don’t know how often approval voting has been tried, or with what results.

IRV has been tested for decades in the real world, and has proved to be a very poor voting method, not much better than plurality (and it degrades to plurality voting with strategic behavior):http://rangevoting.org/AusAboveTheLine07.html

Approval Voting and Range Voting led to 500 years of peace and stability in ancient Venice, only being stopped by Napoleon’s invasion. Approval Voting has also been used for years in several organizations with tens of thousands of members, rivaling the size of many cities.http://rangevoting.org/Approval.html

And the computer simulations show it to be robustly superior to IRV. IRV really is just an incredibly bad method, not fit for human consumption.

Although I feel attracted to approval voting, I have to say that the Gibbard-Satterthwaite is debatable as a theoretical argument in its favor. AV is immune to this theorem only by defining it away. It overtly grants the tactical voting that the theorem says must exist.

Rob Richie’s depraved tactics of basically flat out lying and otherwise being generally misleading don’t convince intelligent people who take their time. Here’s a refutation of his argument against Range Voting:http://rangevoting.org/RichieRV.html

Also, William Poundstone’s new book Gaming the Vote, talks about this, and gets a rave review from Kenneth Arrow himself. The book promotes Range Voting but talks about Approval, Borda, Condorcet, and even IRV. While Poundstone gives Rob Richie fair chance to spew his lying tripe, he allows Princeton math Ph.D. and voting expert Warren D. Smith put up a shield of scientific rigor. Brams gets his shot to defend Approval Voting as well. It’s a great book, especially for politics/voting nerds.

Sigh, rangevoting.org is even more tendentious than fairvote.org. I’d like to say that I agree more with the range voting Mensheviks than with the IRV Bolsheviks, but I can’t. One interesting fact about rangevoting.org is that it is supported by Warren D. Smith. He is a mathematician at Temple with (among other achievements) a very interesting bound on the minimum complexity of a triangulation of the n-cube.

Also, there is one extremely unfortunate version of asset voting used in the United States: campaign contributions.

Range voting is an awful lot better than IRV, but the rangevoting.org web site itself is likely to put people off. Warren Smith may himself be part of the problem – he’s a smart guy, but he can be highly eccentric and he has little tact. I sometimes think he enjoys provoking people, and he at least does a lot of provocative things. I’m convinced his advocacy is a net negative for range voting’s future prospects. By selectively quoting from his opinions and writings, one can make it appear that range voting is supported by nutcases. This isn’t fair, but I hate having to explain to people who are familiar with Smith’s opinions that range voting actually is good.

Gil: Thanks for that link. I’m quite impressed with the system, and I’m convinced that it’s a good system to use when a collective subjective grading is required. However, I’m not convinced that it’s just as good for elections. They seem to be aware of this; their theorem 14 (re)states the impossibility of preventing voters from manipulating overall winners or rankings.

Greg:
Indeed, rangevoting.org’s claims about the Australian electoral system:

IRV has been tested for decades in the real world, and has proved to be a very poor voting method, not much better than plurality (and it degrades to plurality voting with strategic behavior):http://rangevoting.org/AusAboveTheLine07.html

incline me to discount most of the rest of the website; there are certainly problems with Australia’s implementation of preferential voting (IRV, more or less), but they’re mostly technical details that could easily vary between different implementations.

Clay:
Moreover, having just posted about the unappealing personal attacks sometimes seen on physics blogs, I might point out that phrases such as “spew his lying tripe” and “depraved tactics” are neither appropriate nor welcome. My delete finger was getting itchy, but I’m guessing that unless Ben posts again about his plans for world domination through voting reform, we’ll be safe.

Hello. I’m glad to see such eminences looking at this blog.
The website I am a prime layer behind, http://rangevotng.org
has a lot of useful info on range voting and related topics.
(I think) and were the bloggers here to help improve that site,
and/or endorse it, that would be useful.

Rob Richie:
“Systems where lesser choices hurt higher choices have very little track record of success for these reasons.”
–False. This history pagehttp://www.rangevoting.org/OlympHist.html
refutes that claim very dramatically.

Rob Richie:
“Instant runoff voting has a much better history of being used”
–I’m not sure what “better” means, but range voting has had a “longer” history.

Richie: “…and has a long history of working.”
–Well, by “working” Richie means, has a long history of leading to
massive 2-party domination like in Australia IRV seats
(data here: http://rangevoting.org/AustralianPol.html )
and a long history of leading to disturbing paradoxes,
such as, in the last (2007) Australian election cycle, of the 150 IRV house races I spotted at least 9 paradoxes despite having
only incomplete election data to work with:http://rangevoting.org/Aus07.html

Terence Tao:
Tao makes some good points, and I agree with his final
“selecting a system solely because it avoids one type of problem is a little dangerous, as it may unwittingly increase the chance of susceptibility to other sorts of problems. (But either approval or IRV would definitely be superior to a first-past-the-post system.)”

However, Tao is mathematically incorrect in his claim
“in approval voting, one never is induced to actually lie about one’s preferences.”
For counterexamples seehttp://rangevoting.org/RVstrat2.html

Ben Webster falls into the same trap with his:
“it certainly is true that it’s never advantageous to say you prefer candidate A to candidate B, when the reverse is true (which is not by any means a claim IRV can make)”
–now actually there ARE assumptions under which Webster and Tao’s false claims are correct. For example in 3-candidate elctions, or in perfect-info scenarios. Discussion here:http://rangevoting.org/GibbSat.html

and by the way that is also relevant to Greg Kuperberg’s remarks –
range voting partially evades the Gibbard-Satterthwaite impossibility theorem, as is explained there.

Ben Webster:
“I think it also depends a lot on what your personal views on what sort of fairness is important in an election.”
–that’s falling into an unending quagmire because “one’s personal views[about what is] important” is subjective. The way out of that quagmire is the “Bayesian Regret” objective Methodology for comparing voting systems:http://rangevoting.org/BayRegDum.html

Greg Kuperberg:
“Instead of computer simulations, it would be better to do real-life experiments…
Of course in the case of IRV, it’s not a hypothetical, since it is used in Australia and New Zealand. I don’t know how often approval voting has been tried, or with what results.”
–Continuum range voting was used for 500+ years in Sparta
and range3 voting for 500+ years in renaissance Venice,
and for millions of years for Honeybees and some ants.
Were the results good? Well… depends on the meaning of “good,”
but in terms of pure longetivity range is the empirical winner.http://rangevoting.org/OlympHist.html
and linked pages (and linked papers).

IRV is not used in New Zealand (is it)? It certainly
has been used in Australia (the largest use) but in my view unsuccessfully since it led tomassive 2-party domination,
which I consider a massive diminution of democracy (minimal
marketplace) immediately.

Greg Kuperberg:
“IRV has a multiseat generalization that converges to proportional representation is important. Again, I would be interested to see such a generalization of AV, or know that there isn’t … one…. converge to PR as the number of seats goes to infinity?”
–Yes, Approval (and Range) do have such a PR generalization. It is called “reweighted range voting.”
See thishttp://www.rangevoting.org/RRV.htmlhttp://www.rangevoting.org/RRVr.html

–Well, that is enough for now. I’m glad I’m able to contribute to
educating such an elite bunch as yourselves. Kuperberg
in particular seems very quick and he’s thought of a lot of things I thought of but in a lot less time (but I did it first, so I can tell
him the answers :)

Greg Kuperberg:
“Sigh, rangevoting.org is even more tendentious than fairvote.org. I’d like to say that I agree more with the range voting Mensheviks than with the IRV Bolsheviks, but I can’t. One interesting fact about rangevoting.org is that it is supported by Warren D. Smith. He is a mathematician at Temple with (among other achievements) a very interesting bound on the minimum complexity of a triangulation of the n-cube.”

–Hi Greg. I’m not at Temple (but was there once).
Sorry about the “tendentiousness.” It is difficult to make a good
website on this. And perhaps more difficult if you are me.

Would you like to help improve the rangevoting.org web site?

Believe me I’d be happy to accept help. And I find it quite plausible
you’d be better than me in many ways, and even if not “better” the labor by somebody “equal or a bit worse” would still help.

Greg Kuperberg:
“Also, there is one extremely unfortunate version of asset voting used in the United States: campaign contributions.”
–well, that was an amusing quip, but that is not asset voting.
(Also, the delegates method used in the presidential primaries
is not asset voting either – although it might naively seem to be, it
is VERY different. Can you see why?)

Scott Morrison:
“Indeed, rangevoting.org’s claims about the Australian electoral system:
IRV has been tested for decades in the real world, and has proved to be a very poor voting method, not much better than plurality (and it degrades to plurality voting with strategic behavior):http://rangevoting.org/AusAboveTheLine07.html
incline me to discount most of the rest of the website…”

–sorry. I agree that quote was bad. I did not write it.
I serve as kind of editor in chief for the site but sometimes people
slip in contributions that I ought to come down on, but do not.
But actually, I just went to look at this webpage to edit this snippet
to make it better, and it wasn’t there, indicating that I or somebody
else had already edited it away! So you see, we are not
totally out to lunch always :)

Anonymous:
“Warren Smith may himself be part of the problem – he’s a smart guy, but he can be highly eccentric and he has little tact. I sometimes think he enjoys provoking people, and he at least does a lot of provocative things. I’m convinced his advocacy is a net negative for range voting’s future prospects. By selectively quoting from his opinions and writings, one can make it appear that range voting is supported by nutcases. This isn’t fair, but I hate having to explain to people who are familiar with Smith’s opinions that range voting actually is good.”
–well, it is hard to say since I don’t know what Anonymous has in mind. But I repeat: YOU can improve the rangevoting.org website –
push the IMPROVE button. More generally I’d be glad to get
out of this business if I could find somebody better to take it over.
As Anonymous says, there may be plenty of reasons to believe
others would be better. But who, exactly, and will they do it.
That is the problem.

Hmm, that is interesting. I had seen other stuff by them, but not this one, which I guess was not around at the time. I think this is an updated version of something I did see. Let me say, though, that their use of MEDIAN-based range voting,
rather than MEAN, was a mistake. Analysed here:http://www.rangevoting.org/MedianVrange.html

Also, B&L kind of ticked me off by not citing stuff I told them to cite,
etc (their “new” ideas often are not). But they clearly performed
a service with their Orsay studyhttp://www.rangevoting.org/OrsayTable.html

“IRV is not used in New Zealand (is it)?”
STV is used for all district health board elections, and some mayoral and council elections. Since the mayoral elections are single-winner elections, STV reduces to IRV in those cases. For more information (some might say too much information) see my comment #8 above.

The various voting paradoxes come down to the cyclic paradox (what should you do in a world where A > B > C > A?).

I don’t think there’s any easy way around this. In particular, approval voting only gets around it if voters are honest about preferences to their own detriment. Can you imagine a high-stakes two-way race in which the candidate preferred by more voters loses because her supporters honestly indicate a tepid preference? In practice, I guess approval voting would be a nightmare from the tactical voting standpoint.

My conclusion from spending time reading about voting systems (a couple of years ago) is that we should be using a system that is likely to pick a Condorcet winner if there is one (ie., one that will pick a Condorcet winner if it exists and voters choose to enumerate a sufficient number of their choices that this can be seen). Beyond that, it seems like all sensible systems have advantages and disadvantages with respect to each other. So the question becomes, which Condorcet-compliant system is the most “practical” — politically and in terms of implementation?

I believe that the most popular IRV systems are not Condorcet-compliant. This can easily be fixed, and I would argue that it should be before promoting IRV.

To put it another way, I would argue that promoters of IRV should be promoting Condorcet-compliant versions, not the simple, standard version which isn’t.

Comments are closed.

Secret Blogging Seminar

A group blog by 8 recent Berkeley mathematics Ph.D.'s. Commentary on our own research, other mathematics pursuits, and whatever else we feel like writing about on any given day. Sort of like a seminar, but with (even) more rude commentary from the audience.